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+### A Pluto.jl notebook ###
+# v0.20.17
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ d12bda70-894b-11f0-2944-ddded1b41f2c
+begin
+ using LinearAlgebra # I(n), Diagonal
+ using ControlSystems # ss, care, c2d, bode, etc.
+ using OrdinaryDiffEq # CT simulations
+ using Plots # plotting
+ using OSQP, SparseArrays # QP/MPC
+ using Random # for LQG noise simulation
+end
+
+# ╔═╡ 6e282c8b-54e8-4e0a-90a7-25967b8d14f7
+# Topic 2 case1: Single Shooting for double integrator (minimum energy, fixed terminal state)
+# Dynamics: ẋ1 = x2, ẋ2 = u
+# Cost: J = 1/2 ∫ u(t)^2 dt (PMP ⇒ u(t) = c1*t - c2)
+# Boundary: x(0) = [x10, x20], x(T) = [0, 0]
+# Unknowns (shooting params): c = [c1, c2]
+
+let
+ using DifferentialEquations, NLsolve
+
+ # ----- Problem setup -----
+ T = 2.0
+ x10 = 1.0
+ x20 = 0.0
+ x0 = [x10, x20] # initial state
+ xT = [0.0, 0.0] # desired terminal state
+
+ # ----- Forward rollout given c = [c1, c2] -----
+ # Control u(t) = c1*t - c2 (from PMP optimality)
+ function rollout(c; save_traj::Bool=false)
+ c1, c2 = c
+ u(t) = c1*t - c2
+ f!(dx, x, p, t) = begin
+ dx[1] = x[2]
+ dx[2] = u(t)
+ end
+ prob = ODEProblem(f!, x0, (0.0, T))
+ if save_traj
+ sol = solve(prob; reltol=1e-10, abstol=1e-10, saveat=range(0, T; length=401))
+ return sol, [u(t) for t in sol.t]
+ else
+ sol = solve(prob; reltol=1e-10, abstol=1e-10, save_everystep=false)
+ return sol(T) # terminal state only
+ end
+ end
+
+ # ----- Residual for NLsolve: F(c) = x(T; c) - xT -----
+ function residual!(F, c)
+ xT_pred = rollout(c)
+ F[1] = xT_pred[1] - xT[1]
+ F[2] = xT_pred[2] - xT[2]
+ end
+
+ # Initial guess for c = [c1, c2] (you can tweak if needed)
+ c0 = [0.0, 0.0]
+
+ # Solve F(c) = 0 by Newton/Broyden (NLsolve picks a method automatically)
+ sol = nlsolve(residual!, c0; xtol=1e-12, ftol=1e-12)
+ @assert sol.f_converged "Shooting did not converge. Try a different c0."
+ ĉ = sol.zero
+ println("Optimal shooting params (c1, c2) = ", ĉ)
+
+ # ----- Reconstruct full optimal trajectories -----
+ soltraj, utraj = rollout(ĉ; save_traj=true)
+ ts = soltraj.t
+ xs = hcat(soltraj.u...)' # size (N, 2)
+ x1s = xs[:,1]; x2s = xs[:,2]
+
+ # ----- Checks -----
+ println("Boundary check: x(0) = ", xs[1,:], " target = ", x0)
+ println("Boundary check: x(T) = ", xs[end,:], " target = ", xT)
+ # Residual norm at solution (should be ~0)
+ Ffin = zeros(2); residual!(Ffin, ĉ)
+ println("Terminal residual ||F|| = ", norm(Ffin))
+
+ # ----- (Optional) plots -----
+ if @isdefined(Plots)
+ p1 = plot(ts, x1s, lw=2, label="x1(t)", xlabel="t", title="State")
+ plot!(ts, x2s, lw=2, label="x2(t)")
+
+ p2 = plot(ts, utraj, lw=2, label="u(t)", xlabel="t", title="Control")
+
+ plot(p1, p2, layout=(1,2), size=(900,400))
+
+ end
+end
+
+
+# ╔═╡ 59f5b6c7-d211-42c0-b662-377e7ca6a3a5
+# Topic 2 case2: Multiple Shooting for Double Integrator
+# Dynamics: ẋ1 = x2, ẋ2 = u
+# Cost: J = 1/2 ∫ u^2 dt
+# Boundary: x(0) = [1,0], x(T) = [0,0]
+
+let
+ using JuMP, Ipopt
+
+ # ----- problem setup -----
+ T = 2.0
+ M = 20 # number of shooting intervals
+ Δt = T/M
+ x0 = [1.0, 0.0]
+ xT = [0.0, 0.0]
+ umax = 5.0 # control bound
+
+ # ----- build NLP -----
+ model = Model(Ipopt.Optimizer)
+ set_silent(model)
+
+ # decision variables
+ @variables(model, begin
+ x[1:2, 0:M] # state at each node
+ -umax <= u[0:M-1] <= umax # piecewise constant control
+ end)
+
+ # initial condition
+ @constraint(model, x[1,0] == x0[1])
+ @constraint(model, x[2,0] == x0[2])
+
+ # dynamics continuity (Euler)
+ for k in 0:M-1
+ @constraint(model, x[1,k+1] == x[1,k] + Δt * x[2,k])
+ @constraint(model, x[2,k+1] == x[2,k] + Δt * u[k])
+ end
+
+ # terminal condition
+ @constraint(model, x[1,M] == xT[1])
+ @constraint(model, x[2,M] == xT[2])
+
+ # objective: minimize sum of u^2
+ @objective(model, Min, sum(0.5 * Δt * u[k]^2 for k=0:M-1))
+
+ # ----- solve -----
+ optimize!(model)
+ println("Status: ", termination_status(model))
+ println("Objective value: ", objective_value(model))
+
+ # ----- extract solution -----
+ xs = [value.(x[:,k]) for k in 0:M]
+ us = [value(u[k]) for k in 0:M-1]
+ ts = collect(0:Δt:T)
+
+ # check boundary
+ println("x(0) = ", xs[1], " target = ", x0)
+ println("x(T) = ", xs[end], " target = ", xT)
+
+ # ----- plot results -----
+ x1s = [xi[1] for xi in xs]
+ x2s = [xi[2] for xi in xs]
+ p1 = plot(ts, x1s, lw=2, label="x1(t)", xlabel="t", title="State")
+ plot!(ts, x2s, lw=2, label="x2(t)")
+ p2 = plot(ts[1:end-1], us, lw=2, label="u(t)", xlabel="t", title="Control")
+ plot(p1, p2, layout=(1,2), size=(900,400))
+end
+
+
+# ╔═╡ c925a661-0a41-4fa2-a772-e7217e44a818
+# Topic 1 case: analytic example with state+control running cost
+
+let
+
+ # ----- Problem parameters -----
+ T = 2.5 # horizon
+ x0 = 1.5 # initial state
+ q = 1.0 # state weight (>0)
+ r = 0.2 # control weight (>0)
+
+ # ----- Analytic PMP solution -----
+ α = sqrt(q/r)
+
+ # x(t) = A cosh(α t) + B sinh(α t)
+ A = x0
+ B = -x0 * coth(α*T) # from x(T)=0
+
+ x_star(t) = A*cosh(α*t) + B*sinh(α*t)
+ xdot_star(t) = α*(A*sinh(α*t) + B*cosh(α*t))
+
+ u_star(t) = xdot_star(t) # because ẋ = u
+ λ_star(t) = -r * xdot_star(t) # because u = -λ/r
+
+ # ----- Verify PMP conditions on a grid -----
+ ts = range(0, T; length=401)
+
+ # residual trackers (max norms over the grid)
+ max_res_state = 0.0 # |ẋ - ∂H/∂λ|
+ max_res_costate = 0.0 # |λ̇ + ∂H/∂x|
+ max_res_opt = 0.0 # |∂H/∂u|
+
+ for i in 2:length(ts)
+ t_prev, t = ts[i-1], ts[i]
+ dt = t - t_prev
+
+ # Finite differences for derivatives (for numerical verification)
+ xdot_fd = (x_star(t) - x_star(t_prev)) / dt
+ lambdadot_fd = (λ_star(t) - λ_star(t_prev)) / dt
+
+ # Hamiltonian partials
+ dH_dλ = u_star(t) # ∂H/∂λ = u
+ dH_dx = q * x_star(t) # ∂H/∂x = q x
+ dH_du = r * u_star(t) + λ_star(t) # optimality residual
+
+ max_res_state = max(max_res_state, abs(xdot_fd - dH_dλ))
+ max_res_costate = max(max_res_costate, abs(lambdadot_fd + dH_dx))
+ max_res_opt = max(max_res_opt, abs(dH_du))
+ end
+
+ println("PMP residuals over grid (should be near zero):")
+ println(" state eqn (ẋ - ∂H/∂λ): ", max_res_state)
+ println(" costate eqn (λ̇ + ∂H/∂x): ", max_res_costate)
+ println(" optimality (∂H/∂u): ", max_res_opt)
+
+ # Boundary checks
+ println("Boundary check: x(0) = ", x_star(0.0), " (target ", x0, ")")
+ println("Boundary check: x(T) = ", x_star(T), " (target 0.0)")
+
+ # ----- Optional plots -----
+ if @isdefined(Plots)
+ p1 = plot(ts, [x_star.(ts), u_star.(ts), λ_star.(ts)],
+ label = ["x(t)" "u(t)" "λ(t)"], xlabel="t", lw=2)
+ display(p1)
+ end
+
+ # Report a few values
+ for t in (0.0, T/2, T)
+ println("t=$(round(t,digits=2)) x=$(x_star(t)) u=$(u_star(t)) λ=$(λ_star(t))")
+ end
+end
+
+
+# ╔═╡ 8b8d8f05-4de5-48bd-8661-787b86014a87
+# Topic 2 case 3: Multiple Shooting for the nonlinear pendulum (Euler integration)
+# θ̇ = ω, ω̇ = u - sin(θ)
+# Cost:
+# J = Σ 1/2 Δt (qθ θ_k^2 + qω ω_k^2 + r u_k^2) + 1/2 (Qfθ θ_M^2 + Qfω ω_M^2)
+
+let
+
+ # ----- Horizon and grid -----
+ T = 2.5 # horizon
+ M = 60 # number of shooting intervals (nodes = M+1)
+ Δt = T / M
+ ts = collect(0:Δt:T)
+
+ # ----- Initial and (soft) terminal target -----
+ θ0, ω0 = pi, 0.0 # start downward, zero speed
+ # You can keep terminal as soft via Qf*, or add hard constraints x_M = 0.
+
+ # ----- Weights -----
+ qθ, qω, r = 1.0, 0.1, 0.05
+ Qfθ, Qfω = 60.0, 12.0 # terminal weights (bigger → closer to upright rest)
+
+ # ----- Control bound (optional but useful) -----
+ umax = 2.0
+
+ # ----- Build NLP -----
+ model = Model(Ipopt.Optimizer)
+ set_silent(model) # comment this to see Ipopt logs
+
+ # Decision variables:
+ # x[:,k] = [θ_k; ω_k] at node k, u[k] is piecewise constant over [t_k, t_{k+1})
+ @variables(model, begin
+ x[1:2, 0:M]
+ -umax <= u[0:M-1] <= umax
+ end)
+
+ # Initial condition
+ @constraint(model, x[1,0] == θ0)
+ @constraint(model, x[2,0] == ω0)
+
+ # Multiple-shooting continuity (Euler)
+ # θ_{k+1} = θ_k + Δt * ω_k
+ # ω_{k+1} = ω_k + Δt * (u_k - sin(θ_k))
+ for k in 0:M-1
+ @NLconstraint(model, x[1,k+1] == x[1,k] + Δt * x[2,k])
+ @NLconstraint(model, x[2,k+1] == x[2,k] + Δt * (u[k] - sin(x[1,k])))
+ end
+
+ # (Optional) Hard terminal constraints (uncomment to force exact upright rest):
+ # @constraint(model, x[1,M] == 0.0)
+ # @constraint(model, x[2,M] == 0.0)
+
+ # Objective: running cost + terminal cost
+ @NLobjective(model, Min,
+ sum(0.5 * Δt * (qθ * x[1,k]^2 + qω * x[2,k]^2 + r * u[k]^2) for k=0:M-1)
+ + 0.5 * (Qfθ * x[1,M]^2 + Qfω * x[2,M]^2)
+ )
+
+ # ----- Solve -----
+ optimize!(model)
+ println("Status: ", termination_status(model))
+ println("Objective value: ", objective_value(model))
+
+ # ----- Extract solution -----
+ θ = [value(x[1,k]) for k in 0:M]
+ ω = [value(x[2,k]) for k in 0:M]
+ ustar = [value(u[k]) for k in 0:M-1]
+
+ println("θ(0) = ", θ[1], " θ(T) = ", θ[end])
+ println("ω(0) = ", ω[1], " ω(T) = ", ω[end])
+ println("|u|_∞ = ", maximum(abs.(ustar)))
+
+ # ----- Plot -----
+ if @isdefined(Plots)
+ p1 = plot(ts, θ, lw=2, label="θ(t)", xlabel="t", title="Pendulum State")
+ plot!(ts, ω, lw=2, label="ω(t)")
+ p2 = plot(ts[1:end-1], ustar, lw=2, label="u(t)", xlabel="t", title="Control")
+ plot(p1, p2, layout=(1,2), size=(900,400))
+ end
+end
+
+
+# ╔═╡ 78835928-e3c8-4817-97ac-d0ac1ba50c63
+begin
+ # 4-mass chain: x = [q1,q2,q3,q4, q̇1,q̇2,q̇3,q̇4]
+ # u acts on mass 1. Outputs (for visualization): q1, q4.
+ function chain4_msmd(; m = ones(4),
+ k = [800.0, 600.0, 600.0, 900.0, 700.0], # k_wallL, k12, k23, k34, k_wallR
+ c = [12.0, 10.0, 10.0, 15.0, 12.0], # c_wallL, c12, c23, c34, c_wallR
+ u_on = 1)
+
+ n = 4
+ M = Diagonal(m)
+ Kq = zeros(n,n)
+ Cq = zeros(n,n)
+
+ # wall left
+ Kq[1,1] += k[1]; Cq[1,1] += c[1]
+ # internal springs/dampers
+ Kq[1,1] += k[2]; Kq[1,2] -= k[2]; Kq[2,1] -= k[2]; Kq[2,2] += k[2]
+ Cq[1,1] += c[2]; Cq[1,2] -= c[2]; Cq[2,1] -= c[2]; Cq[2,2] += c[2]
+
+ Kq[2,2] += k[3]; Kq[2,3] -= k[3]; Kq[3,2] -= k[3]; Kq[3,3] += k[3]
+ Cq[2,2] += c[3]; Cq[2,3] -= c[3]; Cq[3,2] -= c[3]; Cq[3,3] += c[3]
+
+ Kq[3,3] += k[4]; Kq[3,4] -= k[4]; Kq[4,3] -= k[4]; Kq[4,4] += k[4]
+ Cq[3,3] += c[4]; Cq[3,4] -= c[4]; Cq[4,3] -= c[4]; Cq[4,4] += c[4]
+
+ # wall right
+ Kq[4,4] += k[5]; Cq[4,4] += c[5]
+
+ Z = zeros(n,n)
+ Eye = I(n)
+ A = [ Z Eye;
+ -M\Kq -M\Cq ]
+
+ Bq = zeros(n,1); Bq[u_on] = 1.0
+ B = [ zeros(n,1);
+ M\Bq ]
+
+ Cy = zeros(2,2n); Cy[1,1] = 1.0; Cy[2,4] = 1.0 # q1, q4
+ D = zeros(2,1)
+
+ return ss(A,B,Cy,D) # continuous-time, SISO
+ end
+
+ # Build the plant globally so all later cells can use it
+ sys = chain4_msmd()
+ A,B,Cy,D = sys.A, sys.B, sys.C, sys.D
+ @show size(A), size(B), size(Cy)
+
+end
+
+# ╔═╡ d849f412-8b3c-4ca9-a9dd-80b70cae8dbc
+begin
+ # ---- Cell 3 (fixed) ----
+ # Build G,H for z = [ Wq*S*q ; Wv*v ], H=0 → N = G' H = 0
+ function build_G_weights(; Wq = Diagonal([4.0, 4.0, 4.0, 8.0]),
+ Wv = 0.2I(4))
+ S = [ 1 -1 0 0;
+ 0 1 -1 0;
+ 0 0 1 -1;
+ 0 0 0 1 ]
+ Gq = Wq * S
+ Gv = Wv
+ G = [ Gq zeros(4,4);
+ zeros(4,4) Gv ]
+ H = zeros(size(G,1), 1) # no direct u in z
+ return G, H
+ end
+
+ G, H = build_G_weights()
+ Q = G' * G
+ R = Matrix(0.5I(1))
+ N = zeros(size(A,1), size(B,2))
+
+
+ P = care(A, B, Q, R, N)[1]
+ K = lqr(A, B, Q, R, N)
+
+ @show size(A), size(B), size(K)
+
+ Acl = A - B * K
+ @show real.(eigvals(Acl))
+
+ # CT 仿真
+ x0 = [0.0, 0.0, 0.0, 0.05, 0.0, 0.0, 0.0, 0.0]
+
+ function cl_lqr!(dx, x, p, t)
+ u = (K * x)[1]
+ dx[:] = A * x - B * u
+ end
+
+ Tsim = 8.0
+ sol = solve(ODEProblem(cl_lqr!, x0, (0.0,Tsim)), Tsit5(); abstol=1e-9, reltol=1e-9)
+
+ t = sol.t
+ X = reduce(hcat, sol.u)
+ uL = [(K * X[:,i])[1] for i in eachindex(t)] .* (-1)
+
+ plt1 = plot(t, X[4,:], lw=2, xlabel="t [s]", ylabel="q4 [m]", title="CT LQR: end mass deflection")
+ plt2 = plot(t, uL, lw=2, xlabel="t [s]", ylabel="u [N]", title="CT LQR: control")
+ plot(plt1, plt2, layout=(2,1), size=(800,600))
+
+end
+
+# ╔═╡ 33e04661-b759-40f6-815a-5b0c60a708be
+begin
+ # —— Discretize ——
+ Ts = 0.02
+ sysd = c2d(sys, Ts)
+ Ad, Bd, Cd, Dd = sysd.A, sysd.B, sysd.C, sysd.D
+ n, m = size(Ad,1), size(Bd,2) # n=8, m=1
+
+
+ Qd = Matrix(Q) # 8×8
+ Rd = Matrix(R) # 1×1
+ Nd = zeros(n, m) # 8×1
+ Qf = 50.0I(n)
+
+
+ # X = Acal*x0 + Bcal*U
+ function stack_dyn(Ad, Bd, Np)
+ n, m = size(Ad,1), size(Bd,2)
+ Acal = [Ad^k for k in 1:Np] |> x -> reduce(vcat, x)
+ Bcal = spzeros(n*Np, m*Np)
+ for i in 1:Np, j in 1:i
+ Bcal[(n*(i-1)+1):(n*i), (m*(j-1)+1):(m*j)] = Ad^(i-j) * Bd
+ end
+ return Acal, Bcal
+ end
+
+
+ function make_Qblk(Qd, Qf, Np)
+ n = size(Qd,1)
+ Qblk = spzeros(n*Np, n*Np)
+ for k in 1:Np
+ Qblk[(n*(k-1)+1):(n*k), (n*(k-1)+1):(n*k)] = sparse(k < Np ? Qd : Qf)
+ end
+ return Qblk
+ end
+
+ function make_Rblk(Rd, Np)
+ m = size(Rd,1)
+ Rblk = spzeros(m*Np, m*Np)
+ for k in 1:Np
+ Rblk[(m*(k-1)+1):(m*k), (m*(k-1)+1):(m*k)] = sparse(Rd)
+ end
+ return Rblk
+ end
+
+ function make_Nblk(Nd, Np)
+ n, m = size(Nd)
+ Nblk = spzeros(n*Np, m*Np)
+ for k in 1:(Np-1)
+ Nblk[(n*(k-1)+1):(n*k), (m*(k-1)+1):(m*k)] = sparse(Nd)
+ end
+ return Nblk
+ end
+
+ function condensed_qp_mats(Ad,Bd,Qd,Rd,Nd,Qf,Np)
+ Acal, Bcal = stack_dyn(Ad,Bd,Np)
+ Qblk = make_Qblk(Qd,Qf,Np) # 640×640
+ Rblk = make_Rblk(Rd,Np) # 80×80
+ Nblk = make_Nblk(Nd,Np) # 640×80
+
+ # H 和 f(x0)
+ H = Bcal' * Qblk * Bcal + (Bcal' * Nblk + Nblk' * Bcal) + Rblk
+ f_of = x0 -> (Bcal' * Qblk * Acal + Nblk' * Acal) * x0
+ return H, f_of, Acal, Bcal, Qblk, Rblk, Nblk
+ end
+
+ Np = 80
+ Hqp, f_of, Acal, Bcal, Qblk, Rblk, Nblk = condensed_qp_mats(Ad,Bd,Qd,Rd,Nd,Qf,Np)
+
+ # (640,80), (640,640), (80,80), (640,80)）
+ @show size(Acal), size(Qblk), size(Rblk), size(Nblk)
+ # Hessian (m*Np, m*Np) = (80,80)
+ @show size(Hqp)
+
+end
+
+# ╔═╡ 04291184-92d8-4c52-b4e7-e2d487e5d87b
+begin
+ # ==== Cell 5: Condensed constraints (|u| and |q4|) ====
+
+ # |u_k| ≤ umax；|q4_k| ≤ qmax
+ umax = 15.0
+ qmax = 0.06
+ q4row = zeros(1, n); q4row[4] = 1.0
+
+ """
+ condensed_constraints(Bcal, Acal, n, m, Np; umax, qrow, qmax)
+
+ """
+ function condensed_constraints(Bcal, Acal, n, m, Np; umax=15.0, qrow=zeros(1,n), qmax=0.06)
+ # -umax ≤ u ≤ umax → [ I; -I ] U ≤ [umax; umax]
+ GUu = [ I(m*Np); -I(m*Np) ] # (2mNp) × (mNp)
+ wu = vcat( fill(umax, m*Np), fill(umax, m*Np) )
+ Su = zeros(size(GUu,1), n) # (2mNp) × n
+
+ # |qrow * x_k| ≤ qmax (k = 1..Np)
+ rows = [ (n*(k-1)+1):(n*k) for k in 1:Np ]
+ G_list = Matrix{Float64}[]; w_list = Vector{Float64}[]; S_list = Matrix{Float64}[]
+ for k in 1:Np
+ Ak = Acal[rows[k], :]
+ Bk = Bcal[rows[k], :]
+ # +qmax
+ push!(G_list, qrow * Bk)
+ push!(w_list, [qmax])
+ push!(S_list, qrow * Ak)
+ # -qmax
+ push!(G_list, -qrow * Bk)
+ push!(w_list, [qmax])
+ push!(S_list, -qrow * Ak)
+ end
+ Gs = reduce(vcat, G_list) # (2Np) × (mNp)
+ ws = reduce(vcat, w_list) # (2Np)
+ Ss = reduce(vcat, S_list) # (2Np) × n
+
+ #
+ G = vcat(GUu, Gs) # ((2mNp+2Np) × (mNp))
+ w = vcat(wu, ws) # ((2mNp+2Np))
+ S = vcat(Su, Ss) # ((2mNp+2Np) × n)
+ return G, w, S
+ end
+
+ Gcon, wcon, Scon = condensed_constraints(Bcal, Acal, n, m, Np; umax=umax, qrow=q4row, qmax=qmax)
+
+ # (2mNp+2Np, mNp)、(2mNp+2Np)、(2mNp+2Np, n)）
+ @show size(Gcon), size(wcon), size(Scon);
+
+end
+
+# ╔═╡ cf285f59-d4b2-4935-959b-c8c9c2669570
+function mpc_sim_condensed(Ad,Bd; H, f_of, G, w, S, Ts, Nsteps, x0)
+ n = size(Ad,1)
+ xk = copy(x0)
+
+ tlog = Vector{Float64}(undef, Nsteps)
+ Ulog = Vector{Float64}(undef, Nsteps)
+ q4log = Vector{Float64}(undef, Nsteps)
+ Xlog = Matrix{Float64}(undef, n, Nsteps)
+
+ for k in 1:Nsteps
+ model = Model(OSQP.Optimizer); set_silent(model)
+ @variable(model, U[1:size(H,1)]) # U ∈ ℝ^{m*Np}
+
+
+ Hpsd = sparse(0.5*(H + H') + 1e-9I(size(H,1)))
+ f = f_of(xk) # Vector{Float64}
+
+
+ @objective(model, Min, 0.5 * dot(U, Hpsd * U) + dot(f, U))
+
+ @constraint(model, G * U .<= w .+ S * xk)
+
+ optimize!(model)
+ Ustar = value.(U)
+ u0 = Ustar[1] # receding horizon
+ xk = Ad * xk + Bd * u0
+
+ tlog[k] = (k-1)*Ts
+ Ulog[k] = u0
+ q4log[k] = xk[4]
+ Xlog[:,k] = xk
+ end
+ return (; tlog, Ulog, q4log, Xlog)
+end
+
+# ╔═╡ e84baea8-bd88-4e09-ac0d-83e26b9fa805
+begin
+ # ---- Run MPC and plot results ----
+ Nsteps = Int(round(8.0 / Ts)) # simulate 8 seconds like CT case
+ mpc = mpc_sim_condensed(Ad, Bd;
+ H=Hqp, f_of=f_of, G=Gcon, w=wcon, S=Scon,
+ Ts=Ts, Nsteps=Nsteps, x0=x0)
+
+ # State (tip deflection q4) with constraint lines
+ p1 = plot(mpc.tlog, mpc.q4log, lw=2, label="q4 (MPC)",
+ xlabel="t [s]", ylabel="q4 [m]",
+ title="MPC (QP, condensed): tip deflection")
+ hline!([ qmax], ls=:dash, label="qmax")
+ hline!([-qmax], ls=:dash, label="-qmax")
+
+ # Control with input bounds
+ p2 = plot(mpc.tlog, mpc.Ulog, lw=2, label="u(t)",
+ xlabel="t [s]", ylabel="u [N]",
+ title="MPC control")
+ hline!([ umax], ls=:dash, label="umax")
+ hline!([-umax], ls=:dash, label="-umax")
+
+ plot(p1, p2, layout=(2,1), size=(900,600))
+
+end
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
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+uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
+version = "1.17.1+0"
+
+[[deps.Xorg_libxkbfile_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
+git-tree-sha1 = "e3150c7400c41e207012b41659591f083f3ef795"
+uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
+version = "1.1.3+0"
+
+[[deps.Xorg_xcb_util_cursor_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
+git-tree-sha1 = "c5bf2dad6a03dfef57ea0a170a1fe493601603f2"
+uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
+version = "0.1.5+0"
+
+[[deps.Xorg_xcb_util_image_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
+uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
+git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
+uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_keysyms_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
+uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_renderutil_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
+uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
+version = "0.3.10+0"
+
+[[deps.Xorg_xcb_util_wm_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
+uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
+version = "0.4.2+0"
+
+[[deps.Xorg_xkbcomp_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
+git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
+uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
+version = "1.4.7+0"
+
+[[deps.Xorg_xkeyboard_config_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
+git-tree-sha1 = "00af7ebdc563c9217ecc67776d1bbf037dbcebf4"
+uuid = "33bec58e-1273-512f-9401-5d533626f822"
+version = "2.44.0+0"
+
+[[deps.Xorg_xtrans_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
+uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
+version = "1.6.0+0"
+
+[[deps.Zlib_jll]]
+deps = ["Libdl"]
+uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
+version = "1.2.13+1"
+
+[[deps.Zstd_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
+uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
+version = "1.5.7+1"
+
+[[deps.eudev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
+uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
+version = "3.2.14+0"
+
+[[deps.fzf_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
+uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
+version = "0.61.1+0"
+
+[[deps.libaom_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "4bba74fa59ab0755167ad24f98800fe5d727175b"
+uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
+version = "3.12.1+0"
+
+[[deps.libass_jll]]
+deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6"
+uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
+version = "0.17.4+0"
+
+[[deps.libblastrampoline_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
+version = "5.11.0+0"
+
+[[deps.libdecor_jll]]
+deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
+git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
+uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
+version = "0.2.2+0"
+
+[[deps.libevdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
+uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
+version = "1.13.4+0"
+
+[[deps.libfdk_aac_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.4+0"
+
+[[deps.libinput_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
+git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
+uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
+version = "1.28.1+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.50+0"
+
+[[deps.libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
+git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.8+0"
+
+[[deps.mtdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
+uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
+version = "1.1.7+0"
+
+[[deps.nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+version = "1.59.0+0"
+
+[[deps.oneTBB_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "d5a767a3bb77135a99e433afe0eb14cd7f6914c3"
+uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
+version = "2022.0.0+0"
+
+[[deps.p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+version = "17.4.0+2"
+
+[[deps.x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "10164.0.1+0"
+
+[[deps.x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "4.1.0+0"
+
+[[deps.xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "1.9.2+0"
+"""
+
+# ╔═╡ Cell order:
+# ╠═d12bda70-894b-11f0-2944-ddded1b41f2c
+# ╠═c925a661-0a41-4fa2-a772-e7217e44a818
+# ╠═6e282c8b-54e8-4e0a-90a7-25967b8d14f7
+# ╠═59f5b6c7-d211-42c0-b662-377e7ca6a3a5
+# ╠═8b8d8f05-4de5-48bd-8661-787b86014a87
+# ╠═78835928-e3c8-4817-97ac-d0ac1ba50c63
+# ╠═d849f412-8b3c-4ca9-a9dd-80b70cae8dbc
+# ╠═33e04661-b759-40f6-815a-5b0c60a708be
+# ╠═04291184-92d8-4c52-b4e7-e2d487e5d87b
+# ╠═cf285f59-d4b2-4935-959b-c8c9c2669570
+# ╠═e84baea8-bd88-4e09-ac0d-83e26b9fa805
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002